Consider the C program fragment below which is meant to divide x by y using repeated subtractions. The variables x, y, q and r are all unsigned int.while (r >= y) { r = r - y; q = q + 1; }Which of the following conditions on the variables x, y, q and r before the execution of the fragment will ensure that the loop terminates in a state satisfying the condition x == (y*q + r)?

The loop computes integer division via repeated subtraction where $q$ is the quotient and $r$ is the remainder.

1. Loop Invariant: The relation $x = y \cdot q + r$ must hold before the loop starts and after every iteration:

   • In each iteration, $r$ becomes $r - y$ and $q$ becomes $q + 1$.

   • The new value is $y \cdot (q + 1) + (r - y) = y \cdot q + y + r - y = y \cdot q + r$.

   • To satisfy this invariant initially, setting $q = 0$ and $r = x$ gives $y \cdot 0 + x = x$, which is identically true.

2. Termination: For the loop while (r >= y) to terminate when dealing with unsigned integers:

   • If $y = 0$, $r \ge 0$ is always true, leading to an infinite loop.

   • Therefore, we must have $y > 0$.

Combining both requirements gives the precondition: (q == 0) && (r == x) && (y > 0).

Hence, option C is correct.